DOI: 10.2478/s11533-007-0013-5 Research article CEJM 5(3) 2007 551–580
Strengthened Moser’s conjecture, geometry of Grunsky coefficients and Fredholm eigenvalues∗
Samuel Krushkal
Department of Mathematics, Bar-Ilan University, 52900 Ramat-Gan, Israel and Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152, USA
Received 31 July 2006 ; accepted 3 April, 2007
Abstract: The Grunsky and Teichmuller¨ norms κ(f)andk(f) of a holomorphic univalent function f in a finitely connected domain D ∞with quasiconformal extension to C are related by κ(f) ≤ k(f). In 1985, Jurgen¨ Moser conjectured that any univalent function in the disk Δ∗ = {z : |z| > 1} can be approximated locally uniformly by functions with κ(f) Keywords: quasiconformal, univalent function, Grunsky coefficient inequalities, universal Teichmuller¨ space, subharmonic function, Strebel’s point, Kobayashi metric, generalized Gaussian curvature, holomorphic curvature, Fredholm eigenvalues. MSC (2000): 30C35, 30C62, 32G15,30F60, 32F45, 53A35 1 Introduction and main results 1.1 Grunsky inequalities and Moser’s conjecture The classical Grunsky theorem states that a holomorphic function f(z)=z+const +O(z−1) in a neighborhood U0 of z = ∞ can be extended to a univalent holomorphic function on ∗ To Reiner Kuhnau¨ on his 70th birthday 552 S. Krushkal / Central European Journal of Mathematics 5(3) 2007 551–580 the disk Δ∗ = {z ∈ C = C ∪{∞}: |z| > 1} if and only if its Grunsky coefficients αmn satisfy the inequalities ∞ √ mn αmnxmxn ≤ 1, (1.1) m,n=1 where αmn are defined by ∞ f(z) − f(ζ) log = − α z−mζ−n, (z, ζ) ∈ (Δ∗)2, (1.2) z − ζ mn m,n=1 ∞ 2 2 2 2 x =(xn) runs over the unit sphere S(l )oftheHilbertspacel with x = |xn| ,and 1 the principal branch of the logarithmic function is chosen (cf. [1]). The quantity ∞ √ 2 κ(f):=sup mn αmnxmxn : x =(xn) ∈ S(l ) (1.3) m,n=1 is called the Grunsky constant (or Grunsky norm)off. Let Σ denote the collection of all univalent holomorphic functions −1 ∗ f(z)=z + b0 + b1z + ···:Δ → C \{0}, (1.4) and let Σ(k) be its subset of the functions with k-quasiconformal extensions to the unit {| | } 0 disk Δ = z < 1 so that f(0) = 0. Put Σ = k Σ(k). This collection closely relates to universal Teichmuller¨ space T modelled as a bounded domain in the Banach space B of holomorphic functions in Δ∗ with norm 2 2 ϕ B =sup(|z| − 1) |ϕ(z)|. (1.5) Δ∗ All ϕ ∈ B can be regarded as the Schwarzian derivatives 2 Sf =(f /f ) − (f /f ) /2 of locally univalent holomorphic functions in Δ∗.ThepointsofT represent the functions f ∈ Σ0 whose minimal dilatation μ μ ∗ k(f):=inf{k(w )= μ ∞ : w |∂Δ = f} determines the Teichmuller¨ metric on T.Herewμ denotes a quasiconformal homeomor- phism of C with the Beltrami coefficient μ(z)=∂zw/∂zw, (1.6) and μ ∞ = ess supC |μ(z)| < 1. (1.7) S. Krushkal / Central European Journal of Mathematics 5(3) 2007 551–580 553 Grunsky’s theorem was strengthened for the functions with quasiconformal extensions by several authors. The basic results obtained by Kuhnau,¨ Pommerenke and Zhuravlev are as follows: for any f ∈ Σ0, we have the inequality κ(f) ≤ k(f); (1.8) on the other hand, if a function f ∈ Σ satisfies the inequality κ(f)